Calculus Examples

$6 \sin (x) + \sin (2 x)$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $6 \sin (x) + \sin (2 x)$ with respect to $x$ is $\frac{d}{d x} [6 \sin (x)] + \frac{d}{d x} [\sin (2 x)]$ .

$\frac{d}{d x} [6 \sin (x)] + \frac{d}{d x} [\sin (2 x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 \sin (x)]$ .

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Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sin (x)$ with respect to $x$ is $6 \frac{d}{d x} [\sin (x)]$ .

$6 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\sin (2 x)]$

Step 1.1.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$6 \cos (x) + \frac{d}{d x} [\sin (2 x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [\sin (2 x)]$ .

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Step 1.1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.3.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

$6 \cos (x) + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]$

Step 1.1.3.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$6 \cos (x) + \cos (u) \frac{d}{d x} [2 x]$

Step 1.1.3.1.3

Replace all occurrences of $u$ with $2 x$ .

$6 \cos (x) + \cos (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$6 \cos (x) + \cos (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$6 \cos (x) + \cos (2 x) (2 \cdot 1)$

Step 1.1.3.4

Multiply $2$ by $1$ .

$6 \cos (x) + \cos (2 x) \cdot 2$

Step 1.1.3.5

Move $2$ to the left of $\cos (2 x)$ .

$f' (x) = 6 \cos (x) + 2 \cos (2 x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 \cos (x) + 2 \cos (2 x)$ .

$6 \cos (x) + 2 \cos (2 x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 \cos (x) + 2 \cos (2 x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 \cos (x) + 2 \cos (2 x) = 0$

Step 2.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$6 \cos (x) + 2 (1 - 2 \sin^{2} (x)) = 0$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$6 \cos (x) + 2 \cdot 1 + 2 (- 2 \sin^{2} (x)) = 0$

Step 2.3.1.2

Multiply $2$ by $1$ .

$6 \cos (x) + 2 + 2 (- 2 \sin^{2} (x)) = 0$

Step 2.3.1.3

Multiply $- 2$ by $2$ .

$6 \cos (x) + 2 - 4 \sin^{2} (x) = 0$

Step 2.4

Solve the equation for $x$ .

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Step 2.4.1

Replace the $- 4 \sin^{2} (x)$ with $- 4 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$6 \cos (x) + 2 - 4 (1 - \cos^{2} (x)) = 0$

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

Apply the distributive property.

$6 \cos (x) + 2 - 4 \cdot 1 - 4 (- \cos^{2} (x)) = 0$

Step 2.4.2.2

Multiply $- 4$ by $1$ .

$6 \cos (x) + 2 - 4 - 4 (- \cos^{2} (x)) = 0$

Step 2.4.2.3

Multiply $- 1$ by $- 4$ .

$6 \cos (x) + 2 - 4 + 4 \cos^{2} (x) = 0$

Step 2.4.3

Subtract $4$ from $2$ .

$6 \cos (x) - 2 + 4 \cos^{2} (x) = 0$

Step 2.4.4

Reorder the polynomial.

$4 \cos^{2} (x) + 6 \cos (x) - 2 = 0$

Step 2.4.5

Substitute $u$ for $\cos (x)$ .

$4 {(u)}^{2} + 6 (u) - 2 = 0$

Step 2.4.6

Factor $2$ out of $4 u^{2} + 6 u - 2$ .

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Step 2.4.6.1

Factor $2$ out of $4 u^{2}$ .

$2 (2 u^{2}) + 6 u - 2 = 0$

Step 2.4.6.2

Factor $2$ out of $6 u$ .

$2 (2 u^{2}) + 2 (3 u) - 2 = 0$

Step 2.4.6.3

Factor $2$ out of $- 2$ .

$2 (2 u^{2}) + 2 (3 u) + 2 (- 1) = 0$

Step 2.4.6.4

Factor $2$ out of $2 (2 u^{2}) + 2 (3 u)$ .

$2 (2 u^{2} + 3 u) + 2 (- 1) = 0$

Step 2.4.6.5

Factor $2$ out of $2 (2 u^{2} + 3 u) + 2 (- 1)$ .

$2 (2 u^{2} + 3 u - 1) = 0$

Step 2.4.7

Divide each term in $2 (2 u^{2} + 3 u - 1) = 0$ by $2$ and simplify.

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Step 2.4.7.1

Divide each term in $2 (2 u^{2} + 3 u - 1) = 0$ by $2$ .

$\frac{2 (2 u^{2} + 3 u - 1)}{2} = \frac{0}{2}$

Step 2.4.7.2

Simplify the left side.

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Step 2.4.7.2.1

Cancel the common factor of $2$ .

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Step 2.4.7.2.1.1

Cancel the common factor.

$\frac{2 (2 u^{2} + 3 u - 1)}{2} = \frac{0}{2}$

Step 2.4.7.2.1.2

Divide $2 u^{2} + 3 u - 1$ by $1$ .

$2 u^{2} + 3 u - 1 = \frac{0}{2}$

Step 2.4.7.3

Simplify the right side.

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Step 2.4.7.3.1

Divide $0$ by $2$ .

$2 u^{2} + 3 u - 1 = 0$

Step 2.4.8

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.9

Substitute the values $a = 2$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 2.4.10

Simplify.

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Step 2.4.10.1

Simplify the numerator.

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Step 2.4.10.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 2.4.10.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.4.10.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.4.10.1.2.2

Multiply $- 8$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 2.4.10.1.3

Add $9$ and $8$ .

$u = \frac{- 3 \pm \sqrt{17}}{2 \cdot 2}$

Step 2.4.10.2

Multiply $2$ by $2$ .

$u = \frac{- 3 \pm \sqrt{17}}{4}$

Step 2.4.11

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.11.1

Simplify the numerator.

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Step 2.4.11.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 2.4.11.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.4.11.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.4.11.1.2.2

Multiply $- 8$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 2.4.11.1.3

Add $9$ and $8$ .

$u = \frac{- 3 \pm \sqrt{17}}{2 \cdot 2}$

Step 2.4.11.2

Multiply $2$ by $2$ .

$u = \frac{- 3 \pm \sqrt{17}}{4}$

Step 2.4.11.3

Change the $\pm$ to $+$ .

$u = \frac{- 3 + \sqrt{17}}{4}$

Step 2.4.11.4

Rewrite $- 3$ as $- 1 (3)$ .

$u = \frac{- 1 \cdot 3 + \sqrt{17}}{4}$

Step 2.4.11.5

Factor $- 1$ out of $\sqrt{17}$ .

$u = \frac{- 1 \cdot 3 - 1 (- \sqrt{17})}{4}$

Step 2.4.11.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{17})$ .

$u = \frac{- 1 (3 - \sqrt{17})}{4}$

Step 2.4.11.7

Move the negative in front of the fraction.

$u = - \frac{3 - \sqrt{17}}{4}$

Step 2.4.12

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.12.1

Simplify the numerator.

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Step 2.4.12.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 2.4.12.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.4.12.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.4.12.1.2.2

Multiply $- 8$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 2.4.12.1.3

Add $9$ and $8$ .

$u = \frac{- 3 \pm \sqrt{17}}{2 \cdot 2}$

Step 2.4.12.2

Multiply $2$ by $2$ .

$u = \frac{- 3 \pm \sqrt{17}}{4}$

Step 2.4.12.3

Change the $\pm$ to $-$ .

$u = \frac{- 3 - \sqrt{17}}{4}$

Step 2.4.12.4

Rewrite $- 3$ as $- 1 (3)$ .

$u = \frac{- 1 \cdot 3 - \sqrt{17}}{4}$

Step 2.4.12.5

Factor $- 1$ out of $- \sqrt{17}$ .

$u = \frac{- 1 \cdot 3 - (\sqrt{17})}{4}$

Step 2.4.12.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{17})$ .

$u = \frac{- 1 (3 + \sqrt{17})}{4}$

Step 2.4.12.7

Move the negative in front of the fraction.

$u = - \frac{3 + \sqrt{17}}{4}$

Step 2.4.13

The final answer is the combination of both solutions.

$u = - \frac{3 - \sqrt{17}}{4}, - \frac{3 + \sqrt{17}}{4}$

Step 2.4.14

Substitute $\cos (x)$ for $u$ .

$\cos (x) = - \frac{3 - \sqrt{17}}{4}, - \frac{3 + \sqrt{17}}{4}$

Step 2.4.15

Set up each of the solutions to solve for $x$ .

$\cos (x) = - \frac{3 - \sqrt{17}}{4}$

$\cos (x) = - \frac{3 + \sqrt{17}}{4}$

Step 2.4.16

Solve for $x$ in $\cos (x) = - \frac{3 - \sqrt{17}}{4}$ .

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Step 2.4.16.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{3 - \sqrt{17}}{4})$

Step 2.4.16.2

Simplify the right side.

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Step 2.4.16.2.1

Evaluate $\arccos (- \frac{3 - \sqrt{17}}{4})$ .

$x = 1.28619336$

Step 2.4.16.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 1.28619336$

Step 2.4.16.4

Solve for $x$ .

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Step 2.4.16.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.28619336$

Step 2.4.16.4.2

Simplify $2 (3.14159265) - 1.28619336$ .

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Step 2.4.16.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.28619336$

Step 2.4.16.4.2.2

Subtract $1.28619336$ from $6.2831853$ .

$x = 4.99699194$

Step 2.4.16.5

Find the period of $\cos (x)$ .

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Step 2.4.16.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.4.16.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.4.16.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.4.16.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4.16.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 1.28619336 + 2 π n, 4.99699194 + 2 π n$ , for any integer $n$

Step 2.4.17

Solve for $x$ in $\cos (x) = - \frac{3 + \sqrt{17}}{4}$ .

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Step 2.4.17.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{3 + \sqrt{17}}{4}$ does not fall in this range, there is no solution.

No solution

Step 2.4.18

List all of the solutions.

$x = 1.28619336 + 2 π n, 4.99699194 + 2 π n$ , for any integer $n$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $6 \sin (x) + \sin (2 x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 1.28619336$ .

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Step 4.1.1

Substitute $1.28619336$ for $x$ .

$6 \sin (1.28619336) + \sin (2 (1.28619336))$

Step 4.1.2

Multiply $2$ by $1.28619336$ .

$6 \sin (1.28619336) + \sin (2.57238673)$

Step 4.2

Evaluate at $x = 4.99699194$ .

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Step 4.2.1

Substitute $4.99699194$ for $x$ .

$6 \sin (4.99699194) + \sin (2 (4.99699194))$

Step 4.2.2

Multiply $2$ by $4.99699194$ .

$6 \sin (4.99699194) + \sin (9.99398388)$

Step 4.3

Evaluate at $x = 4.99699194 + 2 π$ .

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Step 4.3.1

Substitute $4.99699194 + 2 π$ for $x$ .

$6 \sin (4.99699194 + 2 π) + \sin (2 (4.99699194 + 2 π))$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Add $4.99699194$ and $2 π$ .

$6 \sin (11.28017724) + \sin (2 (4.99699194 + 2 π))$

Step 4.3.2.2

Add $4.99699194$ and $2 π$ .

$6 \sin (11.28017724) + \sin (2 \cdot 11.28017724)$

Step 4.3.2.3

Multiply $2$ by $11.28017724$ .

$6 \sin (11.28017724) + \sin (22.56035449)$

Step 4.4

Evaluate at $x = 4.99699194 + 4 π$ .

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Step 4.4.1

Substitute $4.99699194 + 4 π$ for $x$ .

$6 \sin (4.99699194 + 4 π) + \sin (2 (4.99699194 + 4 π))$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Add $4.99699194$ and $4 π$ .

$6 \sin (17.56336255) + \sin (2 (4.99699194 + 4 π))$

Step 4.4.2.2

Add $4.99699194$ and $4 π$ .

$6 \sin (17.56336255) + \sin (2 \cdot 17.56336255)$

Step 4.4.2.3

Multiply $2$ by $17.56336255$ .

$6 \sin (17.56336255) + \sin (35.12672511)$

Step 4.5

Evaluate at $x = 4.99699194 + 6 π$ .

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Step 4.5.1

Substitute $4.99699194 + 6 π$ for $x$ .

$6 \sin (4.99699194 + 6 π) + \sin (2 (4.99699194 + 6 π))$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Add $4.99699194$ and $6 π$ .

$6 \sin (23.84654786) + \sin (2 (4.99699194 + 6 π))$

Step 4.5.2.2

Add $4.99699194$ and $6 π$ .

$6 \sin (23.84654786) + \sin (2 \cdot 23.84654786)$

Step 4.5.2.3

Multiply $2$ by $23.84654786$ .

$6 \sin (23.84654786) + \sin (47.69309572)$

Step 4.6

Evaluate at $x = 4.99699194 + 8 π$ .

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Step 4.6.1

Substitute $4.99699194 + 8 π$ for $x$ .

$6 \sin (4.99699194 + 8 π) + \sin (2 (4.99699194 + 8 π))$

Step 4.6.2

Simplify each term.

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Step 4.6.2.1

Add $4.99699194$ and $8 π$ .

$6 \sin (30.12973317) + \sin (2 (4.99699194 + 8 π))$

Step 4.6.2.2

Add $4.99699194$ and $8 π$ .

$6 \sin (30.12973317) + \sin (2 \cdot 30.12973317)$

Step 4.6.2.3

Multiply $2$ by $30.12973317$ .

$6 \sin (30.12973317) + \sin (60.25946634)$

Step 4.7

List all of the points.

$(1.28619336 + 2 π n, 6 \sin (1.28619336) + \sin (2.57238673)), (4.99699194, 6 \sin (4.99699194) + \sin (9.99398388)), (4.99699194 + 2 π, 6 \sin (11.28017724) + \sin (22.56035449)), (4.99699194 + 4 π, 6 \sin (17.56336255) + \sin (35.12672511)), (4.99699194 + 6 π, 6 \sin (23.84654786) + \sin (47.69309572)), (4.99699194 + 8 π, 6 \sin (30.12973317) + \sin (60.25946634))$ , for any integer $n$

Step 5